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Properties

Label 12-7098e6-1.1-c1e6-0-1
Degree $12$
Conductor $1.279\times 10^{23}$
Sign $1$
Analytic cond. $3.31496\times 10^{10}$
Root an. cond. $7.52846$
Motivic weight $1$
Arithmetic yes
Rational yes
Primitive no
Self-dual yes
Analytic rank $0$

Origins

Origins of factors

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Normalization:  

Dirichlet series

L(s)  = 1  + 6·2-s + 6·3-s + 21·4-s + 9·5-s + 36·6-s − 6·7-s + 56·8-s + 21·9-s + 54·10-s + 10·11-s + 126·12-s − 36·14-s + 54·15-s + 126·16-s + 4·17-s + 126·18-s + 2·19-s + 189·20-s − 36·21-s + 60·22-s + 336·24-s + 30·25-s + 56·27-s − 126·28-s − 4·29-s + 324·30-s + 9·31-s + ⋯
L(s)  = 1  + 4.24·2-s + 3.46·3-s + 21/2·4-s + 4.02·5-s + 14.6·6-s − 2.26·7-s + 19.7·8-s + 7·9-s + 17.0·10-s + 3.01·11-s + 36.3·12-s − 9.62·14-s + 13.9·15-s + 63/2·16-s + 0.970·17-s + 29.6·18-s + 0.458·19-s + 42.2·20-s − 7.85·21-s + 12.7·22-s + 68.5·24-s + 6·25-s + 10.7·27-s − 23.8·28-s − 0.742·29-s + 59.1·30-s + 1.61·31-s + ⋯

Functional equation

\[\begin{aligned}\Lambda(s)=\mathstrut &\left(2^{6} \cdot 3^{6} \cdot 7^{6} \cdot 13^{12}\right)^{s/2} \, \Gamma_{\C}(s)^{6} \, L(s)\cr=\mathstrut & \,\Lambda(2-s)\end{aligned}\]
\[\begin{aligned}\Lambda(s)=\mathstrut &\left(2^{6} \cdot 3^{6} \cdot 7^{6} \cdot 13^{12}\right)^{s/2} \, \Gamma_{\C}(s+1/2)^{6} \, L(s)\cr=\mathstrut & \,\Lambda(1-s)\end{aligned}\]

Invariants

Degree: \(12\)
Conductor: \(2^{6} \cdot 3^{6} \cdot 7^{6} \cdot 13^{12}\)
Sign: $1$
Analytic conductor: \(3.31496\times 10^{10}\)
Root analytic conductor: \(7.52846\)
Motivic weight: \(1\)
Rational: yes
Arithmetic: yes
Character: Trivial
Primitive: no
Self-dual: yes
Analytic rank: \(0\)
Selberg data: \((12,\ 2^{6} \cdot 3^{6} \cdot 7^{6} \cdot 13^{12} ,\ ( \ : [1/2]^{6} ),\ 1 )\)

Particular Values

\(L(1)\) \(\approx\) \(12409.30654\)
\(L(\frac12)\) \(\approx\) \(12409.30654\)
\(L(\frac{3}{2})\) not available
\(L(1)\) not available

Euler product

   \(L(s) = \displaystyle \prod_{p} F_p(p^{-s})^{-1} \)
$p$$F_p(T)$
bad2 \( ( 1 - T )^{6} \)
3 \( ( 1 - T )^{6} \)
7 \( ( 1 + T )^{6} \)
13 \( 1 \)
good5 \( 1 - 9 T + 51 T^{2} - 216 T^{3} + 746 T^{4} - 426 p T^{5} + 5173 T^{6} - 426 p^{2} T^{7} + 746 p^{2} T^{8} - 216 p^{3} T^{9} + 51 p^{4} T^{10} - 9 p^{5} T^{11} + p^{6} T^{12} \)
11 \( 1 - 10 T + 57 T^{2} - 254 T^{3} + 1077 T^{4} - 3962 T^{5} + 13499 T^{6} - 3962 p T^{7} + 1077 p^{2} T^{8} - 254 p^{3} T^{9} + 57 p^{4} T^{10} - 10 p^{5} T^{11} + p^{6} T^{12} \)
17 \( 1 - 4 T + 47 T^{2} - 84 T^{3} + 1047 T^{4} - 990 T^{5} + 18595 T^{6} - 990 p T^{7} + 1047 p^{2} T^{8} - 84 p^{3} T^{9} + 47 p^{4} T^{10} - 4 p^{5} T^{11} + p^{6} T^{12} \)
19 \( 1 - 2 T + 89 T^{2} - 164 T^{3} + 3631 T^{4} - 5684 T^{5} + 87437 T^{6} - 5684 p T^{7} + 3631 p^{2} T^{8} - 164 p^{3} T^{9} + 89 p^{4} T^{10} - 2 p^{5} T^{11} + p^{6} T^{12} \)
23 \( 1 + 71 T^{2} + 14 T^{3} + 2705 T^{4} - 476 T^{5} + 74193 T^{6} - 476 p T^{7} + 2705 p^{2} T^{8} + 14 p^{3} T^{9} + 71 p^{4} T^{10} + p^{6} T^{12} \)
29 \( 1 + 4 T + 147 T^{2} + 513 T^{3} + 9571 T^{4} + 27987 T^{5} + 356706 T^{6} + 27987 p T^{7} + 9571 p^{2} T^{8} + 513 p^{3} T^{9} + 147 p^{4} T^{10} + 4 p^{5} T^{11} + p^{6} T^{12} \)
31 \( 1 - 9 T + 91 T^{2} - 550 T^{3} + 3836 T^{4} - 20192 T^{5} + 117065 T^{6} - 20192 p T^{7} + 3836 p^{2} T^{8} - 550 p^{3} T^{9} + 91 p^{4} T^{10} - 9 p^{5} T^{11} + p^{6} T^{12} \)
37 \( 1 - 9 T + 229 T^{2} - 1516 T^{3} + 21046 T^{4} - 106070 T^{5} + 1031497 T^{6} - 106070 p T^{7} + 21046 p^{2} T^{8} - 1516 p^{3} T^{9} + 229 p^{4} T^{10} - 9 p^{5} T^{11} + p^{6} T^{12} \)
41 \( 1 - 25 T + 337 T^{2} - 2764 T^{3} + 14176 T^{4} - 36436 T^{5} + 56273 T^{6} - 36436 p T^{7} + 14176 p^{2} T^{8} - 2764 p^{3} T^{9} + 337 p^{4} T^{10} - 25 p^{5} T^{11} + p^{6} T^{12} \)
43 \( 1 - 19 T + 6 p T^{2} - 2330 T^{3} + 18425 T^{4} - 118803 T^{5} + 798208 T^{6} - 118803 p T^{7} + 18425 p^{2} T^{8} - 2330 p^{3} T^{9} + 6 p^{5} T^{10} - 19 p^{5} T^{11} + p^{6} T^{12} \)
47 \( 1 - 21 T + 387 T^{2} - 4550 T^{3} + 48661 T^{4} - 401373 T^{5} + 3071622 T^{6} - 401373 p T^{7} + 48661 p^{2} T^{8} - 4550 p^{3} T^{9} + 387 p^{4} T^{10} - 21 p^{5} T^{11} + p^{6} T^{12} \)
53 \( 1 + 4 T + 165 T^{2} + 573 T^{3} + 15485 T^{4} + 48151 T^{5} + 1018922 T^{6} + 48151 p T^{7} + 15485 p^{2} T^{8} + 573 p^{3} T^{9} + 165 p^{4} T^{10} + 4 p^{5} T^{11} + p^{6} T^{12} \)
59 \( 1 - 20 T + 263 T^{2} - 2493 T^{3} + 15927 T^{4} - 77885 T^{5} + 461594 T^{6} - 77885 p T^{7} + 15927 p^{2} T^{8} - 2493 p^{3} T^{9} + 263 p^{4} T^{10} - 20 p^{5} T^{11} + p^{6} T^{12} \)
61 \( 1 - 3 T + 255 T^{2} - 688 T^{3} + 31017 T^{4} - 78425 T^{5} + 2347614 T^{6} - 78425 p T^{7} + 31017 p^{2} T^{8} - 688 p^{3} T^{9} + 255 p^{4} T^{10} - 3 p^{5} T^{11} + p^{6} T^{12} \)
67 \( 1 - 24 T + 409 T^{2} - 4773 T^{3} + 46461 T^{4} - 394593 T^{5} + 3242986 T^{6} - 394593 p T^{7} + 46461 p^{2} T^{8} - 4773 p^{3} T^{9} + 409 p^{4} T^{10} - 24 p^{5} T^{11} + p^{6} T^{12} \)
71 \( 1 - 13 T + 182 T^{2} - 1590 T^{3} + 19893 T^{4} - 179473 T^{5} + 1909328 T^{6} - 179473 p T^{7} + 19893 p^{2} T^{8} - 1590 p^{3} T^{9} + 182 p^{4} T^{10} - 13 p^{5} T^{11} + p^{6} T^{12} \)
73 \( 1 + 9 T + 340 T^{2} + 1638 T^{3} + 42975 T^{4} + 100641 T^{5} + 3413872 T^{6} + 100641 p T^{7} + 42975 p^{2} T^{8} + 1638 p^{3} T^{9} + 340 p^{4} T^{10} + 9 p^{5} T^{11} + p^{6} T^{12} \)
79 \( 1 - 28 T + 689 T^{2} - 10269 T^{3} + 141835 T^{4} - 1459913 T^{5} + 14626910 T^{6} - 1459913 p T^{7} + 141835 p^{2} T^{8} - 10269 p^{3} T^{9} + 689 p^{4} T^{10} - 28 p^{5} T^{11} + p^{6} T^{12} \)
83 \( 1 - 15 T + 336 T^{2} - 3546 T^{3} + 46725 T^{4} - 416711 T^{5} + 4377468 T^{6} - 416711 p T^{7} + 46725 p^{2} T^{8} - 3546 p^{3} T^{9} + 336 p^{4} T^{10} - 15 p^{5} T^{11} + p^{6} T^{12} \)
89 \( 1 - 11 T + 515 T^{2} - 4532 T^{3} + 111908 T^{4} - 776666 T^{5} + 13172095 T^{6} - 776666 p T^{7} + 111908 p^{2} T^{8} - 4532 p^{3} T^{9} + 515 p^{4} T^{10} - 11 p^{5} T^{11} + p^{6} T^{12} \)
97 \( 1 - 2 T + 239 T^{2} - 821 T^{3} + 34987 T^{4} - 150177 T^{5} + 3546130 T^{6} - 150177 p T^{7} + 34987 p^{2} T^{8} - 821 p^{3} T^{9} + 239 p^{4} T^{10} - 2 p^{5} T^{11} + p^{6} T^{12} \)
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   \(L(s) = \displaystyle\prod_p \ \prod_{j=1}^{12} (1 - \alpha_{j,p}\, p^{-s})^{-1}\)

Imaginary part of the first few zeros on the critical line

−4.06706228683932828761466409068, −3.87092400634502753372252638835, −3.66743422576824851457417048142, −3.65362674821261649718243930590, −3.55761491304720856301936887872, −3.49726240312064182435805967125, −3.43345157880337896734305520570, −2.98503447213174522029603590702, −2.80249656002224272462207406963, −2.73155250695659942741855752207, −2.69149674666955871297456724433, −2.61862683708070889566841884890, −2.60799469974861787268775393815, −2.27316688174740262660715746612, −2.21301652545314418201939533376, −2.17412360446382019235418423953, −1.94502446403246636822821965644, −1.93127574320986776694076065244, −1.88856771373797201133937172433, −1.30245989499711008107689654340, −1.16933600708609646345686514696, −1.11992803314556897558068449719, −0.969899334244049287901954485480, −0.874436884400245323763826412027, −0.806747181628321201250786682576, 0.806747181628321201250786682576, 0.874436884400245323763826412027, 0.969899334244049287901954485480, 1.11992803314556897558068449719, 1.16933600708609646345686514696, 1.30245989499711008107689654340, 1.88856771373797201133937172433, 1.93127574320986776694076065244, 1.94502446403246636822821965644, 2.17412360446382019235418423953, 2.21301652545314418201939533376, 2.27316688174740262660715746612, 2.60799469974861787268775393815, 2.61862683708070889566841884890, 2.69149674666955871297456724433, 2.73155250695659942741855752207, 2.80249656002224272462207406963, 2.98503447213174522029603590702, 3.43345157880337896734305520570, 3.49726240312064182435805967125, 3.55761491304720856301936887872, 3.65362674821261649718243930590, 3.66743422576824851457417048142, 3.87092400634502753372252638835, 4.06706228683932828761466409068

Graph of the $Z$-function along the critical line

Plot not available for L-functions of degree greater than 10.